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On the M-eigenvalue estimation of fourth-order partially symmetric tensors
1. | School of Mathematics and Information Science, Weifang University, Weifang Shandong, 261061, China |
2. | School of Management Science, Qufu Normal University, Rizhao Shandong, 276826, China |
In this article, the M-eigenvalue of fourth-order partially symmetric tensors is estimated by choosing different components of M-eigenvector. As an application, some upper bounds for the M-spectral radius of nonnegative fourth-order partially symmetric tensors are discussed, which are sharper than existing upper bounds. Finally, numerical examples are reported to verify the obtained results.
References:
[1] |
K. Chang, K. Pearson and T. Zhang,
Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.
doi: 10.1016/j.laa.2013.02.013. |
[2] |
H. Chen, Z. Huang and L. Qi,
Copositivity detection of tensors: Theory and algorithm, J. Optimiz. Theory Appl., 174 (2017), 746-761.
doi: 10.1007/s10957-017-1131-2. |
[3] |
H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Linear Algebra Appl., 25 (2018), e2125, 16pp.
doi: 10.1002/nla.2125. |
[4] |
H. Chen, Z. Huang and L. Qi,
Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69 (2018), 133-158.
doi: 10.1007/s10589-017-9938-1. |
[5] |
H. Chen and Y. Wang,
On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China, 12 (2017), 1289-1302.
doi: 10.1007/s11464-017-0645-0. |
[6] |
H. Chen, L. Qi and Y. Song,
Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.
doi: 10.1007/s11464-018-0681-4. |
[7] |
S. Chirit$\check{a}$, A. Danescu and M. Ciarletta,
On the srtong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.
doi: 10.1007/s10659-006-9096-7. |
[8] |
B. Dacorogna,
Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Cont. Dyn-B, 1 (2001), 257-263.
doi: 10.3934/dcdsb.2001.1.257. |
[9] |
W. Ding, J. Liu, L. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, preprint, arXiv: 1705.09911. |
[10] |
D. Han, H. Dai and L. Qi,
Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.
doi: 10.1007/s10659-009-9205-5. |
[11] |
J. He and T. Huang,
Upper bound for the largest Z-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.
doi: 10.1016/j.aml.2014.07.012. |
[12] |
Z. Huang and L. Qi,
Positive definiteness of paired symmetric tensors and elasticity tensors, J. Comput. Appl. Math., 338 (2018), 22-43.
doi: 10.1016/j.cam.2018.01.025. |
[13] |
J. K. Knowles and E. Sternberg,
On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elasticity, 5 (1975), 341-361.
doi: 10.1007/BF00126996. |
[14] |
J. K. Knowles and E. Sternberg,
On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal., 63 (1977), 321-336.
doi: 10.1007/BF00279991. |
[15] |
E. Kofidis and P. Regalia,
On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.
doi: 10.1137/S0895479801387413. |
[16] |
C. Padovani,
Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.
doi: 10.1023/A:1020946506754. |
[17] |
L. Qi, H. Dai and D. Han,
Conditions for strong ellipticity and M-eigenvalues, Front. Math. China, 4 (2009), 349-364.
doi: 10.1007/s11464-009-0016-6. |
[18] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[19] |
Y. Song and L. Qi,
Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.
doi: 10.1137/130909135. |
[20] |
J. R. Walton and J. P. Wilber,
Sufficient conditions for strong ellipticity for a class of anisotropic materials, Int. J. Nonlin. Mech., 38 (2003), 441-455.
doi: 10.1016/S0020-7462(01)00066-X. |
[21] |
Y. Wang, L. Caccetta and G. Zhou,
Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra and Appl., 22 (2015), 1059-1076.
doi: 10.1002/nla.1996. |
[22] |
X. Wang, H. Chen and Y. Wang,
Solution structures of tensor complementarity problem, Front. Math. China, 13 (2018), 935-945.
doi: 10.1007/s11464-018-0675-2. |
[23] |
Y. Wang, L. Qi and X. Zhang,
A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.
doi: 10.1002/nla.633. |
[24] |
Y. Wang, K. Zhang and H. Sun,
Criteria for strong H-tensors, Front. Mathe. China, 11 (2016), 577-592.
doi: 10.1007/s11464-016-0525-z. |
[25] |
G. Wang, G. Zhou and L. Caccetta,
Z-eigenvalue inclusion theorems for tensors, Discrete Cont. Dyn-B, 22 (2017), 187-198.
doi: 10.3934/dcdsb.2017009. |
[26] |
T. Zhang and G. Golub,
Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.
doi: 10.1137/S0895479899352045. |
[27] |
K. Zhang and Y. Wang,
An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math., 305 (2016), 1-10.
doi: 10.1016/j.cam.2016.03.025. |
[28] |
G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134, 10pp.
doi: 10.1002/nla.2134. |
show all references
References:
[1] |
K. Chang, K. Pearson and T. Zhang,
Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.
doi: 10.1016/j.laa.2013.02.013. |
[2] |
H. Chen, Z. Huang and L. Qi,
Copositivity detection of tensors: Theory and algorithm, J. Optimiz. Theory Appl., 174 (2017), 746-761.
doi: 10.1007/s10957-017-1131-2. |
[3] |
H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Linear Algebra Appl., 25 (2018), e2125, 16pp.
doi: 10.1002/nla.2125. |
[4] |
H. Chen, Z. Huang and L. Qi,
Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69 (2018), 133-158.
doi: 10.1007/s10589-017-9938-1. |
[5] |
H. Chen and Y. Wang,
On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China, 12 (2017), 1289-1302.
doi: 10.1007/s11464-017-0645-0. |
[6] |
H. Chen, L. Qi and Y. Song,
Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.
doi: 10.1007/s11464-018-0681-4. |
[7] |
S. Chirit$\check{a}$, A. Danescu and M. Ciarletta,
On the srtong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.
doi: 10.1007/s10659-006-9096-7. |
[8] |
B. Dacorogna,
Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Cont. Dyn-B, 1 (2001), 257-263.
doi: 10.3934/dcdsb.2001.1.257. |
[9] |
W. Ding, J. Liu, L. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, preprint, arXiv: 1705.09911. |
[10] |
D. Han, H. Dai and L. Qi,
Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.
doi: 10.1007/s10659-009-9205-5. |
[11] |
J. He and T. Huang,
Upper bound for the largest Z-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.
doi: 10.1016/j.aml.2014.07.012. |
[12] |
Z. Huang and L. Qi,
Positive definiteness of paired symmetric tensors and elasticity tensors, J. Comput. Appl. Math., 338 (2018), 22-43.
doi: 10.1016/j.cam.2018.01.025. |
[13] |
J. K. Knowles and E. Sternberg,
On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elasticity, 5 (1975), 341-361.
doi: 10.1007/BF00126996. |
[14] |
J. K. Knowles and E. Sternberg,
On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal., 63 (1977), 321-336.
doi: 10.1007/BF00279991. |
[15] |
E. Kofidis and P. Regalia,
On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.
doi: 10.1137/S0895479801387413. |
[16] |
C. Padovani,
Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.
doi: 10.1023/A:1020946506754. |
[17] |
L. Qi, H. Dai and D. Han,
Conditions for strong ellipticity and M-eigenvalues, Front. Math. China, 4 (2009), 349-364.
doi: 10.1007/s11464-009-0016-6. |
[18] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[19] |
Y. Song and L. Qi,
Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.
doi: 10.1137/130909135. |
[20] |
J. R. Walton and J. P. Wilber,
Sufficient conditions for strong ellipticity for a class of anisotropic materials, Int. J. Nonlin. Mech., 38 (2003), 441-455.
doi: 10.1016/S0020-7462(01)00066-X. |
[21] |
Y. Wang, L. Caccetta and G. Zhou,
Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra and Appl., 22 (2015), 1059-1076.
doi: 10.1002/nla.1996. |
[22] |
X. Wang, H. Chen and Y. Wang,
Solution structures of tensor complementarity problem, Front. Math. China, 13 (2018), 935-945.
doi: 10.1007/s11464-018-0675-2. |
[23] |
Y. Wang, L. Qi and X. Zhang,
A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.
doi: 10.1002/nla.633. |
[24] |
Y. Wang, K. Zhang and H. Sun,
Criteria for strong H-tensors, Front. Mathe. China, 11 (2016), 577-592.
doi: 10.1007/s11464-016-0525-z. |
[25] |
G. Wang, G. Zhou and L. Caccetta,
Z-eigenvalue inclusion theorems for tensors, Discrete Cont. Dyn-B, 22 (2017), 187-198.
doi: 10.3934/dcdsb.2017009. |
[26] |
T. Zhang and G. Golub,
Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.
doi: 10.1137/S0895479899352045. |
[27] |
K. Zhang and Y. Wang,
An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math., 305 (2016), 1-10.
doi: 10.1016/j.cam.2016.03.025. |
[28] |
G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134, 10pp.
doi: 10.1002/nla.2134. |



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